Sparse quantum error-correcting codes obey the simple constraint that only a constant
number of qubits need to be measured at a time to extract syndrome bits. Considerable effort
has been devoted to studying sparse quantum codes, most notably in the context of
topological quantum error correction [1]}. This effort is driven by the fact that the
sparsity constraint is quite natural physically, and existing fault-tolerant
thresholds [2]} and overheads [3]} are optimized when
the underlying code is sparse. Despite this effort, finding families of good sparse
quantum codes – i.e. codes with asymptotically constant rate and relative distance – remains
an open problem, in stark contrast to the situation for classical codes
(see e.g. [4]}).
